A graph $H$ is defined if the number of monochromatic copies of $H$ in the two-edge coloring of the full graph $K_n$ is asymptotically minimized by a random coloring, or equivalently $t_H(W )+t_H( 1-W)\geq 2^{1-e(H)}$ holds for all graphones $W.[0,1]^2\right arrow [0,1]$, where $t_H(.)$ represents the homomorphic density of the graph $H$. The generality of paths and cycles is one of the early foundations of extreme graph theory by Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989).
We prove graph homomorphic inequalities that extend the commonality of paths and cycles. That is, $t_H(W)+t_H(1-W)\geq t_{K_2}(W)^{e(H)} +t_{K_2}(1-W)^{e(H)}$ anytime $ H$ is the path or cycle and $W:[0,1]^2\rightarrow\mathbb{R}$ is a bounded symmetric measurable function.
This answers Sidorenko’s question in 1989, who proved that even-length path results are slightly weaker to prove commonality for odd cycles. Moreover, asking whether inequalities hold for graphone $W$ and odd-cycle $H$, he solves the recent conjectures of Behague, Morrison, and Noel in a powerful form. Our proof uses the Schur convexity of fully homogeneous symmetric functions.
